Theorem mircgrextend 28935

Description: Link congruence over a pair of mirror points. cf tgcgrextend 28730. (Contributed by Thierry Arnoux, 4-Oct-2020.)

Hypotheses

Ref	Expression
mirval.p	⊢ 𝑃 = (Base‘𝐺)
mirval.d	⊢ − = (dist‘𝐺)
mirval.i	⊢ 𝐼 = (Itv‘𝐺)
mirval.l	⊢ 𝐿 = (LineG‘𝐺)
mirval.s	⊢ 𝑆 = (pInvG‘𝐺)
mirval.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
mirtrcgr.e	⊢ ∼ = (cgrG‘𝐺)
mirtrcgr.m	⊢ 𝑀 = (𝑆‘𝐵)
mirtrcgr.n	⊢ 𝑁 = (𝑆‘𝑌)
mirtrcgr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
mirtrcgr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
mirtrcgr.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
mirtrcgr.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
mircgrextend.1	⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))

Assertion

Ref	Expression
mircgrextend	⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Proof of Theorem mircgrextend

Step	Hyp	Ref	Expression
1		mirval.p	. 2 ⊢ 𝑃 = (Base‘𝐺)
2		mirval.d	. 2 ⊢ − = (dist‘𝐺)
3		mirval.i	. 2 ⊢ 𝐼 = (Itv‘𝐺)
4		mirval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5		mirtrcgr.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
6		mirtrcgr.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
7		mirval.l	. . 3 ⊢ 𝐿 = (LineG‘𝐺)
8		mirval.s	. . 3 ⊢ 𝑆 = (pInvG‘𝐺)
9		mirtrcgr.m	. . 3 ⊢ 𝑀 = (𝑆‘𝐵)
10	1, 2, 3, 7, 8, 4, 6, 9, 5	mircl 28914	. 2 ⊢ (𝜑 → (𝑀‘𝐴) ∈ 𝑃)
11		mirtrcgr.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
12		mirtrcgr.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13		mirtrcgr.n	. . 3 ⊢ 𝑁 = (𝑆‘𝑌)
14	1, 2, 3, 7, 8, 4, 12, 13, 11	mircl 28914	. 2 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝑃)
15	1, 2, 3, 7, 8, 4, 6, 9, 5	mirbtwn 28911	. . 3 ⊢ (𝜑 → 𝐵 ∈ ((𝑀‘𝐴)𝐼𝐴))
16	1, 2, 3, 4, 10, 6, 5, 15	tgbtwncom 28733	. 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼(𝑀‘𝐴)))
17	1, 2, 3, 7, 8, 4, 12, 13, 11	mirbtwn 28911	. . 3 ⊢ (𝜑 → 𝑌 ∈ ((𝑁‘𝑋)𝐼𝑋))
18	1, 2, 3, 4, 14, 12, 11, 17	tgbtwncom 28733	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼(𝑁‘𝑋)))
19		mircgrextend.1	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝑋 − 𝑌))
20	1, 2, 3, 4, 5, 6, 11, 12, 19	tgcgrcomlr 28725	. . 3 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝑌 − 𝑋))
21	1, 2, 3, 7, 8, 4, 6, 9, 5	mircgr 28910	. . 3 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝐵 − 𝐴))
22	1, 2, 3, 7, 8, 4, 12, 13, 11	mircgr 28910	. . 3 ⊢ (𝜑 → (𝑌 − (𝑁‘𝑋)) = (𝑌 − 𝑋))
23	20, 21, 22	3eqtr4d 2806	. 2 ⊢ (𝜑 → (𝐵 − (𝑀‘𝐴)) = (𝑌 − (𝑁‘𝑋)))
24	1, 2, 3, 4, 5, 6, 10, 11, 12, 14, 16, 18, 19, 23	tgcgrextend 28730	1 ⊢ (𝜑 → (𝐴 − (𝑀‘𝐴)) = (𝑋 − (𝑁‘𝑋)))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  distcds 17318  TarskiGcstrkg 28672  Itvcitv 28678  LineGclng 28679  cgrGccgrg 28755  pInvGcmir 28905

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-trkgc 28693  df-trkgb 28694  df-trkgcb 28695  df-trkg 28698  df-mir 28906

This theorem is referenced by:  mirtrcgr  28936